Characterization of Global Fields by Dirichlet L-Series

Cornelissen et al. Res. Number Theory (2019) 5:7 https://doi.org/10.1007/s40993-018-0143-9

RESEARCH Characterization of global ﬁelds by Dirichlet L-series Gunther Cornelissen1,BartdeSmit2,XinLi3, Matilde Marcolli4,5,6 and Harry Smit1*

*Correspondence: [email protected] Abstract 1Mathematisch Instituut, Universiteit Utrecht, Postbus We prove that two global fields are isomorphic if and only if there is an isomorphism of 80.010, 3508 TA Utrecht, The groups of Dirichlet characters that preserves L-series. Netherlands Keywords: L Full list of author information is Class field theory, -series, Arithmetic equivalence available at the end of the article Mathematics Subject Classification: 11R37, 11R42, 11R56, 14H30 Part of this work was done whilst the first and third author enjoyed the hospitality of the University of Warwick (special thanks to Richard Sharp for making it 1 Introduction possible). As was discovered by Gaßmann in 1926 [10], number fields are not uniquely determined up to isomorphism by their zeta functions. A theorem of Tate [20] from 1966 implies the same for global function fields. At the other end of the spectrum, results of Neukirch and Uchida [15,21,22] around 1970 state that the absolute Galois group does uniquely determine a global field. The better understood abelianized Galois group again does not determine the field up to isomorphism at all, as follows from the description of its character group by Kubota in 1957 ([13], compare [1,17]). Funakura ([9,§I],using[8, Thm. 5]) has shown in 1980 that there exists a number field k (of degree 255! over Q) and two non- isomorphic abelian extensions K/k and L/k of degree 4 with a bijection between all Artin L-series of K/k and L/k (or, equivalently, between the L-series of the four occurring characters). In this paper, we prove that two global fields K and L are isomorphic if and only if there q q ab ∼ q ab exists an isomorphism of groups of Dirichlet characters ψ : GK −→ GL that preserves q q ab L-series: LK(χ) = LL(ψ(χ)) for all χ ∈ GK . A more detailed series of equivalences can be found in the Main Theorem 3.1 below. To connect this theorem to the discussion in the previous paragraph, observe that the existence of ψq without equality of L-series is the same as K and L having abelianized Galois groups that are isomorphic as topological

groups, and that for the trivial character χtriv,wehaveLK(χtriv) = ζK, so that preserving L-series at χtriv is the same as K and L having the same zeta function. Contrary to the result of Funakura, our characters do not factor over a fixed Galois group. In global function fields we explicitly construct the isomorphism of function fields via a map of kernels of reciprocity maps (much akin to the final step in Uchida’s proof [22]). For number fields, we do not need the full hypothesis: we can prove the stronger result that for every number field, there exists a character of any chosen order > 2 for which the L-series does not equal any other Dirichlet L-series of any other field (see Theorem 10.1).

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The method here is not via the kernel of the reciprocity map, but rather via representation theory. We briefly indicate the relation of our work to previous results. The main result was first stated in a 2010 preprint by two of the current authors, who discovered it through methods from mathematical physics ([4], now split into two parts [5]and[6]). For function fields, the main result was proven by one of the authors in [3], using dynamical systems, referring, however, to [4] for some auxiliary results, of which we present the first published proofs in the current paper. Sections 9 and 10 contain an independent proof for the number field case that was found by the second author in 2011. The current paper not only presents full and simplified proofs, but also uses only classical methods from number theory (class field theory, Chebotarev, Grunwald-Wang, and inverse Galois theory). In [19], reconstruction of field extensions of a fixed rational function field from relative L-series is treated from the point of view of the method in Sects. 9 and 10 of this paper. After some preliminaries in the first section, we state the main result in Sect. 3.The next few sections outline the proof of the various equivalences in the main theorem, using basic class field theory and work of Uchida and Hoshi. In the final sections, we deal with number fields, and use representation theory to prove some stronger results.

2 Preliminaries In this section, we set notation and introduce the main object of study. A monoid is a semigroup with identity element. If R is a ring, we let R∗ denote its group of invertible elements. Given a global field K,weusethewordprime to denote a non-zero prime ideal if K is a number field, and to denote an irreducible effective divisor if K is a global function field. Let PK be the set of primes of K.Ifp ∈ PK,letvp be the normalized (additive) valuation

corresponding to p, Kp the local field at p,andOp its ring of integers. Let AK,f be the finite ∗ K OK A adele ring of , its ring of finite integral adeles and K,f the group of ideles (invertible finite adeles), all with their usual topology. Note that in the function field case, infinite placesdonotexist,sothatinthatcase,AK,f = AK is the adele ring, OK is the ring of A∗ = A∗ K integral adeles and K,f K is the full group of ideles. If is a number field, we denote by OK its ring of integers. If K is a global function field, we denote by q the cardinality of the constant field. Let IK be the multiplicative monoid of non-zero integral ideals/effective divisors of our K P m ∈ global field ,soIK is generated by K. We extend the valuation to ideals: if IK p ∈ P m ∈ Z m = pvp(m) and K,wedefinevp( ) ≥0 by requiring p∈P K .LetN be the norm function on the monoid IK: it is the multiplicative function defined on primes p by p N(p):= #OK/p if K is a number field and N(p) = qdeg( ) if K is a function field. Given two global fields K and L, we call a monoid homomorphism ϕ : IK → IL norm-preserving if N(ϕ(m)) = N(m) for all m ∈ IK. We have a surjective monoid homomorphism · A∗ ∩ O → → pvp(xp) ( )K : K,f K IK, (xp)p , p a restriction of the usual homomorphism from ideles to fractional ideals—we use the monoid version in Sect. 6.1 and in the companion paper [6]. A split for (·)K is by definition ∗ K K → A ∩ OK p K p = a monoid homomorphism s : I K,f such that for every prime , s ( ) ... π ... π ∈ K · ◦ = ( , 1, p, 1, ) for some uniformizer p p. It follows that ( )K sK idIK . Cornelissen et al. Res. Number Theory (2019) 5:7 Page 3 of 15 7

ab ab Let GK be the Galois group of a maximal abelian extension K of K, a profinite topo- ∗ ab logical group. There is an Artin reciprocity map AK → GK . In the number field case, we A∗ A∗ A∗ → ∈ A∗ embed K,f into the group of ideles K via K,f x (1,x) K, restrict the Artin ∗ A K K reciprocity map to K,f and call this restriction rec . In the function field case, rec is just the (full) Artin reciprocity map. q ab ab Let T denote the unit circle, equipped with the usual topology, and GK = Hom (GK , T) ab q ab the group of continuous linear characters of GK . Given χ ∈ GK ,wewrite χ = p ∈ P χ| ∗ = U( ): K: recK(Op) 1 χ χ for the set of primes where is unramified .Wedenoteby U( ) the submonoid of IK χ m ∈ χ m = p ∈ P \ χ generated by U( ), i.e., IK is in U( ) if and only if vp( ) 0 for all K U( ). ∗ For m ∈ U(χ) , we set (in a well-defined way, as the choice of sK is up to an element of OK):

χ(m):= χ(recK(sK(m))), and for m ∈ IK\ U(χ) we set χ(m) = 0. q ab ab For any χ ∈ GK , the kernel ker χ is an open subgroup of GK . Therefore, the fixed field of χ, denoted Kχ , is a finite abelian extension of K. Even more: as the extension is finite, there is an n such that χ n is the trivial character. It follows that im χ is a subgroup of ∼ the nth roots of unity, hence cyclic. As we have an isomorphism im χ −→ Gal(Kχ /K), we

obtain that Kχ /K is a ﬁnite cyclic extension. Conversely, for any ﬁnite cyclic extension K q ab ab of K there exists a character χ ∈ GK such that ker χ = Gal(K /K ). The following two lemmas are easy, but we include a proof in the terminology of this paper for lack of a suitable reference.

q ab Lemma 2.1 Let χ ∈ GK . The primes in U(χ) are exactly the primes that are unramiﬁed in Kχ .

ab Proof The character χ factors through the quotient map GK Gal(Kχ /K), giving an ∗ injective character χ :Gal(Kχ /K) → T. Under the quotient map, the group recK(Op) ∗ is mapped surjectively to the inertia group Ip(Kχ /K), hence we have χ(recK(Op)) = χ(Ip(Kχ /K)). As χ is injective, this group is equal to {1} precisely when the inertia group is trivial, i.e. when p is unramiﬁed in Kχ .

∗ Lemma 2.2 For any prime p ∈ PK,setNp := ker χ.ThenrecK(Op) = Np,and χ: p∈U(χ) the associated fixed field is equal to Kur,p, the maximal abelian extension of K unramified at p.

q ab ∗ Proof By definition of U(χ), for any χ ∈ GK with p ∈ U(χ)wehaverecK(Op) ⊆ ker χ, ∗ ∗ ∗ hence recK(Op) ⊆ Np. As Op is compact and recK is continuous, recK(Op) is compact, ab ∗ and as GK is Hausdorff, recK(Op)isclosed. ∗ K ∗ O ab Let Op denote the fixed field of recK( p). Under the quotient map GK ∗ K ∗ /K O K ∗ /K Gal( Op ), recK( p) is mapped to the inertia group Ip( Op ) by class field theory, but it is also mapped to {1} by definition. Hence the inertia group is trivial, and

O ∗ p Kab recK( p) = K ∗ ⊆ Kur, ( ) Op .

The extension associated to Np contains the composite of the extensions Kχ associated to the individual χ. The field Kur,p is a composite of finite cyclic extensions of K unramified 7 Page 4 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

at p. As mentioned, for any such a ﬁnite cyclic extension K there exists a χ such that ker χ = Gal(Kab/K ). Because this extension is unramiﬁed at p, the previous lemma asserts that p ∈ U(χ). It follows that p Kur, ⊆ (Kab)Np . ∗ Combining these two results, we obtain recK(Op) ⊇ Np, and the result follows.

The (Dirichlet) L-function attached to a character χ of K is the function of the complex variable s: −s − −s LK(χ) = (1 − χ(p)N(p) ) 1 = χ(m)N(m) . p∈U(χ) m∈U(χ) (We drop the ﬁeld K from the notation for the L-series unless confusion might arise.) Let χ be a character with associated extension Kχ . For the primes in U(χ)wehave that recK(sK(p)) mod ker χ is independent of the choice of the split sK, and equal to the

Frobenius Frobp in Gal(K /K); note that in a general extension, “the” Frobenius of an unramiﬁed prime in K is a conjugacy class in the Galois group, but in our setting of abelian extensions, it is an actual element. The L-series can thus be written as − − L(χ) = (1 − χ(Frobp)N(p) s) 1. p∈U(χ)

3 The main theorem Theorem 3.1 Let K and L be two global ﬁelds. The following are equivalent:

(i) There exists ∼ • a monoid isomorphism ϕ : IK −→ IL, ab ∼ ab • an isomorphism of topological groups ψ : GK −→ GL ,and ∗ ∗ • K K → A ∩ OK L L → A ∩ OL splits s : I K,f ,s : I L,f such that ψ O∗ = O∗ p K (recK( p)) recL( ϕ(p)) for every prime of , (1) ψ(recK(sK(m))) = recL(sL(ϕ(m))) for all m ∈ IK. (2)

(ii) There exists ∼ • a norm-preserving monoid isomorphism ϕ : IK −→ IL,and ab ∼ ab • an isomorphism of topological groups ψ : GK −→ GL

N such that for every finite abelian extension K = Kab of K (N a subgroup in Gab) K ψ(N) with corresponding field extension L = Lab of L, ϕ is a bijection between the unramified primes of K /K and L /L such that

ψ(Frobp) = Frobϕ(p) in Gal(L /L).

ab ∼ ab (iii) There exists an isomorphism of topological groups ψ : GK −→ GL such that

q q ab L(χ) = L(ψ(χ)) for all χ ∈ GK , (3)

where ψq is given by ψq(χ) = χ ◦ ψ−1. (iv) K and L are isomorphic as ﬁelds. Cornelissen et al. Res. Number Theory (2019) 5:7 Page 5 of 15 7

We will refer to condition (i) as a reciprocity isomorphism, condition (ii) as a finite reciprocity isomorphism, and condition (iii) as an L-isomorphism. We will prove the implication (i) ⇒ (ii) in Proposition 4.2, implication (ii) ⇒ (iii) in Proposition 5.1, and implication (iii) ⇒ (i) in Proposition 6.1. The equivalence with (iv) is proven in 8.1 for function fields and follows from 10.1 for number fields.

4 From reciprocity isomorphism to ﬁnite reciprocity isomorphism Proposition 4.1 Assume 3.1(i).Thenϕ is norm-preserving, i.e.

N(p) = N(ϕ(p)) for all p ∈ PK. (4)

Proof We have

O∗ −→∼ O∗ −→∼ O∗ −→∼ O∗ p recK( p) recL( ϕ(p)) ϕ(p)

for all p ∈ PK as the local reciprocity map is injective. Thus, it suffices to find a way to ∗ read off N(p) from the isomorphism type of Op. Let tors(A) be the torsion subgroup of an abelian group A,andletAp be the p-primary part of a finite abelian group A. Assume that N(p) = pf for some prime p and f ≥ 1. The following facts follow from [16, Chapter II, Proposition (5.7)]:

∗ ∗ p ∗ ∗ • p is uniquely determined by the property Op/tors(Op) = Op/tors(Op), f ∗ ∗ • N(p) = p =|tors(Op)/tors(Op)p|+1.

So we indeed have (4).

Proposition 4.2 Condition 3.1(i) implies condition 3.1(ii).

∗ Proof Aprimep of K is unramiﬁed in K ifandonlyifrecK(Op) ⊆ N. There- fore, (1) implies that ϕ(p) is unramiﬁed in L , and the bijection follows by symmetry. Also, recK(sK(p)) mod N is independent of the choice of the split sK, and equal to the

Frobenius Frobp in Gal(K /K). Hence from (2) we obtain ψ(Frobp) = Frobϕ(p). This proves (ii).

5 From ﬁnite reciprocity isomorphism to L-isomorphism q q ab Proposition 5.1 Assume 3.1(ii).ThenL(χ) = L(ψ(χ)) for all χ ∈ GK .

q ab ab ker χ Proof Let χ ∈ GK ,andletKχ = (K ) denote the ﬁxed ﬁeld of ker χ. As mentioned in Sect. 2, the associated L-series can be written as − − L(χ) = (1 − χ(Frobp)N(p) s) 1. p∈U(χ)

ψ χ = ψq χ ψq L = Lab ψ(ker χ) We have (ker ) ker( ( )) by deﬁnition of .Let ψq(χ) ( ) .Let q p ∈ PK and q := ϕ(p) ∈ PL. By assumption, ϕ(U(χ)) = U(ψ(χ)), N(p) = N(q) q and if p ∈ U(χ) then ψ(Frobp) = Frobq and thus χ(Frobp) = ψ(χ)(Frobq). It follows that 7 Page 6 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

− − L(χ) = (1 − χ(Frobp)N(p) s) 1 p∈ χ U( ) q − − = (1 − ψ(χ)(Frobq)N(q) s) 1 q∈U(ψq(χ)) = L(ψq(χ)).

6 From L-isomorphism to reciprocity isomorphism Proposition 6.1 Assume that there exists an isomorphism of topological groups ψ : ab ∼ ab q q ab GK −→ GL with L(χ) = L(ψ(χ)) for all χ ∈ GK .Then3.1(i) holds.

Proof The idea of the proof is to construct a bijection of primes ϕ with the desired properties via the use of a type of characters that have value 1 on all but one of the primes of a certain norm. For the proof, which will occupy this entire section, we will use the following notation. Whenever an object has subscript N, < N,or≥ N, all associated sets of (prime) ideals are

restricted to (prime) ideals of norm N, < N or ≥ N respectively. For example, UN (χ)is the set of primes of norm N at which χ is unramiﬁed. We use the multiplicative form of the L-series, i.e. −s −1 L

p∈U

and L≥N (χ) is deﬁned similarly.

Our approach is as follows: for every N ∈ N we construct a bijection

ϕN : PK,N → PL,N q q ab such that χ(p) = ψ(χ)(ϕN (p)) for all χ ∈ GK and all p ∈ PK,N . That suﬃces to prove (i), as we now ﬁrst explain. We obtain a bijection

ϕ : PK → PL ϕ| = ϕ ϕ χ p = ψq χ ϕ p χ ∈ q ab by setting P K,N : N , so that satisﬁes ( ) ( )( ( )), for all GK and all p ∈ PK. From this, the following sequence of equivalences follows: p ∈ U(χ) ⇐⇒ χ(p) = 0 ⇐⇒ ψq(χ)(ϕ(p)) = 0 ⇐⇒ ϕ(p) ∈ U(ψq(χ)). Because ϕ is a bijection, we obtain ϕ(U(χ)) = U(ψq(χ)). Certainly ϕ gives rise to a monoid ∼ isomorphism IK −→ IL,and(1) follows from Lemma 2.2 combined with ϕ(U(χ)) = q q q ab U(ψ(χ)). Thus, for every p ∈ PK, χ(p) = ψ(χ)(ϕ(p)) for all χ ∈ GK with p ∈ U(χ) implies that for any splits sK and sL, ψ p ≡ ϕ p O∗ (recK(sK( ))) recL(sL( ( ))) mod recL( ϕ(p)). Hence we can modify our splits to obtain (2), and this proves (i).

To construct ϕN , the strategy is to proceed inductively on N.ForN = 1, the statement is empty. Assume that we have constructed ϕM for all M < N.Let

be the norm-preserving bijection obtained by combining the ϕM.Since q q U

it follows that q L

We can apply this equality to prove the following lemma.

q ab Lemma 6.2 For any character χ ∈ GK , deﬁne

XN (χ):= χ(p) = χ(p). p∈P K,N p∈UN (χ)

q q ab Then XN (χ) = XN (ψ(χ)) for all χ ∈ GK . As a result, |PK,N |=|PL,N |. Proof Let PK,≥N be the submonoid of IK generated by PK,≥N . We write L≥N (χ)in additive form: ⎛ ⎞

−s ⎝ ⎠ −s L≥N (χ) = χ(m)N(m) = χ(m) M . ≥ m∈P K,≥N M N m∈P K,≥N ∩IK,M

q −s As L≥N (χ) = L≥N (ψ(χ)), the coeﬃcients of N in both sums are equal, hence

χ(m) = ψq(χ)(n).

m∈P K,≥N ∩IK,N n∈P L,≥N ∩IL,N As PK,≥N ∩ IK,N = PK,N and PL,≥N ∩ IL,N = PL,N , the desired equality follows. The ﬁnal result is obtained by setting χ = 1.

q ab Deﬁnition 6.3 Let cN :=|PK,N |=|PL,N |. For any character χ ∈ GK ,deﬁne

• uN (χ) =|UN (χ)|, • VN (χ) ={p ∈ UN (χ) | χ(p) = 1},and • vN (χ) =|VN (χ)|.

If vN (χ) = uN (χ) − 1, there exists a unique prime of norm N on which χ has a value of neither 0 nor 1. We will denote this prime by pχ . Lastly, we deﬁne the following sets of characters: 1 q ab K := χ ∈ GK : uN (χ) = vN (χ) = cN 2 q ab K := χ ∈ GK : uN (χ) = cN ,vN (χ) = cN − 1, χ(pχ ) = ζ ,

where ζ = exp(2πi/k) for some ﬁxed integer k ≥ 3.

q ab Remark 6.4 Let χ ∈ GK .As|χ(p)|=1 for all p ∈ UN (χ), we have Re(XN (χ)) ≤ uN (χ). Equality holds precisely when χ(p) = 1 for all p ∈ UN (χ), i.e. uN (χ) = vN (χ).

q 1 1 Lemma 6.5 ψ(K) = L.

1 q Proof From Lemma 6.2 we obtain χ ∈ K ⇐⇒ XN (χ) = cN ⇐⇒ XN (ψ(χ)) = q 1 cN ⇐⇒ ψ(χ) ∈ L. 7 Page 8 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

Lemma 6.6 For k ≥ 3, let μk denote the kth roots of unity and let ζ = exp(2πi/k). Supposewehavea1, ...,an ∈ μk ∪{0} such that a1 +···+an = n − 1 + ζ . Then there is a j such that aj = ζ ,andai = 1 for all i = j.

Proof If k = 3, then ζ is the only possible value with a positive imaginary part. Hence

one of the aj equals ζ , and then ai = 1 for the remaining i = j, since they have to sum to n − 1. For k > 3, let R := max Re(μk −{1}), and let f denote the number of ai that are not equal to 1. Since 0 ≤ R = Re(ζ ) < 1, we ﬁnd that

n − 1 + R = n − 1 + Re(ζ ) = Re(a1 +···+an) ≤ n − f + f Re(ζ ) = n + f (R − 1),

so R − 1 ≤ f (R − 1), hence f ≤ 1.

q 2 2 Lemma 6.7 ψ(K) = L.

2 k 1 q k 1 Proof For any character χ ∈ K we have χ ∈ K. Thus, by Lemma 6.5, ψ(χ) ∈ L. q q Hence for any q ∈ PL,N we have that ψ(χ)(q) ∈ μk ∪{0}.AsXN (ψ(χ)) = XN (χ) = − +ζ q ψq χ q = cN 1 , by the previous lemma there exists a single prime ψq(χ) such that ( )( ψq(χ)) ζ ψq χ q = q = q ψq 2 ⊆ 2 , while ( )( ) 1 for all ψq(χ). Hence ( K) L. By symmetry, we have equality.

2 A character χ in K has a special prime pχ of norm N, and the corresponding character ψq χ ∈ 2 q ( ) L has a special prime ψq(χ) of norm N. We obtain a relation between primes

2 { pχ q χ ∈ }⊆PK × PL ( , ψq(χ)): K ,N ,N

which we will now show is the graph {(pχ , ϕN (pχ )} of a bijection ϕN : PK,N → PL,N with the required property. The ﬁrst step is to show that every prime of PK,N is associated to at least one prime of PL,N .

2 Lemma 6.8 For every p ∈ PK,N there exists a character χ ∈ K such that pχ = p .

Proof The Grunwald-Wang Theorem [2, Ch. X, Thm. 5] guarantees that there exists a q ab character χ ∈ GK such that χ(p ) = ζ and χ(p) = 1 for all primes p = p of norm N, because there exists a character of Gab whose fixed field is the unique unramified Kp extension of degree k of Kp .

Lemma 6.9 ϕ PK → PL pχ → q The map N : ,N ,N : ψq(χ) is a well-deﬁned bijection such q ab q that for every χ ∈ GK and p ∈ PK,N we have χ(p) = ψ(χ)(ϕN (p)).

2 χ χ ∈ pχ = p q = q Proof Suppose we have , K such that χ and ψq(χ) ψq(χ ).Wehave 2 XN (χ · χ ) = cN − 1 + ζ ,

while

q q q 2 XN (ψ(χ · χ )) = XN (ψ(χ) · ψ(χ )) = cN − 2 + 2ζ = cN − 1 + ζ , q = q ψq −1 ψq which contradicts Lemma 6.2. We conclude that ψq(χ) ψq(χ ).Using instead of ϕ−1 provides a well-deﬁned inverse N . Cornelissen et al. Res. Number Theory (2019) 5:7 Page 9 of 15 7

2 Take χ ∈ K such that pχ = p.Wehave

XN (χ · χ ) = XN (χ) − χ(p) + ζχ(p) = XN (χ) − (1 − ζ )χ(p). Similarly, q q q q q XN (ψ(χ · χ )) = XN (ψ(χ) · ψ(χ )) = XN (ψ(χ)) − (1 − ζ )ψ(χ)(ϕN (p)). q q As we have both XN (χ) = XN (ψ(χ)) and XN (χ · χ ) = XN (ψ(χ · χ )), we conclude q q that (1 − ζ )χ(p) = (1 − ζ )ψ(χ)(ϕN (p)) and consequently χ(p) = ψ(χ)(ϕN (p)).

This completes the proof of Proposition 6.1.

7 Conditional reconstruction of global ﬁelds Proposition 7.1 Assume the equivalent statements (i)–(iii) of Theorem 3.1. Then there exists an isomorphism such that

∗ ∗ A ∩ OK A ∩ OL K,f L,f (5)

recK recL ψ ab ab GK GL

commutes.

∗ ∗ O × K → A ∩ OK m → · K m Proof Deﬁne the homomorphism K I K,f by (u, ) u s ( ). It has a ∗ ∗ − A ∩ OK → O × K → · K K 1 K complete inverse K,f K I given by x (x s ((x) ) , (x) ). We obtain ∗ ∼ ∗ A ∩ OK −→ O × K L an isomorphism K,f K I (and similarly for ). As seen in the proof of Proposition 4.1, ψ O∗ −→∼ O∗ −→∼ O∗ −→∼ O∗ p recK( p) recL( ϕ(p)) ϕ(p). We obtain an isomorphism

O∗ −→∼ O∗ p : p ϕ(p) (6)

that ﬁts into the following commutative diagram: O∗ p O∗ p ϕ(p)

recK recL ψ ab ab GK GL ψ m = ϕ m m ∈ As we have (recK(sK( ))) recL(sL( ( ))) for all IK by assumption, the map ( p) × ϕ ﬁts in the commutative diagram ×ϕ ∗ ( p) ∗ OK × IK OL × IL

recK recL ψ ab ab GK GL ∗ ∼ ∗ ∗ ∼ ∗ A ∩OK −→ O × K A ∩OL −→ O × L Using the aforementioned identiﬁcations K,f K I and L,f L I we obtain the desired isomorphism . 7 Page 10 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

Corollary 7.2 Diagram (5) can be extended to a commutative diagram

A∗ A∗ K,f L,f (7)

recK recL ψ ab ab GK GL

A∗ ab Proof The reciprocity map is already deﬁned from K,f to GK , so this follows from (5) ∗ ∗ A A ∩ OK by passing to the group of fractions K,f of the monoid K,f .

We now turn to the reconstruction of isomorphism of fields from the equivalent conditions in the Main Theorem 3.1. For this, we first quote a result about conditions under which an isomorphism of multiplicative groups of fields can be extended to a field iso-

morphism. Let O[p] denote the local ring (of K) at the prime p.

Lemma 7.3 (Uchida/Hoshi) An isomorphism F : K∗ → L∗ of multiplicative groups of two global ﬁelds K and L is the restriction of an isomorphism of ﬁelds if and only if there exists a bijection ϕ : PK → PL such that for all p ∈ PK, both the following hold:

(i) F(1 + pO[p]) = 1 + ϕ(p)O[ϕ(p)] (as sets, or, equivalently, as subgroups), (ii) vϕ(p) ◦ F = vp.

Proof This follows immediately by results of Uchida for global function ﬁelds [22, Lemma 8–11], as explained in the introduction of [12]) and by Hoshi for number ﬁelds [12, Thm. D].

Theorem 7.4 Assume that in (7) above satisﬁes (K∗) = L∗. Then the extension of that isomorphism to a map K → L by setting 0 → 0 is an isomorphism of ﬁelds.

∗ ∗ p ∈ PK p A K Proof Fix .Let : K,f p be the canonical projection, which we use to K∗ A∗ K∗ project elements of (that are diagonally embedded in K,f ) into the completions p. ∗ ∼ ∗ p K −→ L By construction, there is an isomorphism : p ϕ(p) such that

ϕ(p) ◦ = p ◦ p.

The one-units of the complete local ring are simply the (N(p) − 1)-th powers [16,Ch. II, Prop. 5.7]:

∗ N(p)−1 1 + πpOp = (Op) . (8)

Since p is multiplicative, and respects units by Eq. (6)weﬁnd

p(1 + pOp) = 1 + ϕ(p)Oϕ(p). ∗ ∗ Setting F := |K∗ : K → L , we conclude that for the local rings, we have

∗ −1 F(1 + pO[p]) = (K ∩ p (1 + pOp)) ∗ − = (K ) ∩ ◦ 1 (1 + pOp) p − = 1 ◦p ϕ(p) = L∗ ∩ −1 + ϕ p O ϕ(p)(1 ( ) ϕ(p)) Cornelissen et al. Res. Number Theory (2019) 5:7 Page 11 of 15 7

= 1 + ϕ(p)O[ϕ(p)]. This proves condition (i) in Lemma 7.3. For condition (ii), observe that from the definition of it follows that (sK(p)) = sL(ϕ(p)). Since p(sK(p)) = πp by definition of a split (for a chosen uniformizer πp), we find that

p(πp) = φ(p)(sL(ϕ(p))) =: πϕ(p) is a uniformizer at ϕ(p). Thus, the multiplicative map F respects elements of valuation 0 (see again Eq. (6)) and 1, and hence (ii) holds.

8 Reconstruction of global function ﬁelds In function ﬁelds, we can immediately apply the results of the previous section:

Proposition 8.1 If one of the equivalent conditions (i)–(iii) of Theorem 3.1 holds for two global function ﬁelds K and L, then they are isomorphic as ﬁelds.

Proof From the commutativity of diagram (7) we ﬁnd that (ker(recK)) = ker(recL). For ∗ a global function ﬁeld K we have ker(recK) = K and therefore the result follows from Theorem 7.4.

Remark 8.2 The equivalence of (iii) in Theorem 3.1 and ﬁeld isomorphism in global function ﬁelds was also shown in [3] using dynamical systems, but referring to the unpublished [4] for a proof of the result in Sect. 6 of this paper.

9 The number field case: an auxiliary result In this section we prove the existence of certain Galois extensions of number fields with prescribed Galois groups; a result that we will use in the next section to prove the reconstruction of number fields from the consideration of specific induced representations.

Proposition 9.1 Let K be a number field of degree n contained in a finite Galois extension N of Q, and let C be a finite cyclic group. Denote G = Gal(N/Q) and H = Gal(N/K) and let Cn G be the semidirect product of Cn and G, where the action of G on Cn is by permuting coordinates the same way G permutes the cosets G/H. By Cn Hwedenote the subgroup of Cn G generated by Cn and H. There exists a Galois extension M of Q containing N such that

Gal(M/Q) = Cn G, Gal(M/K) = Cn H, and Gal(M/N) = Cn.

Remark 9.2 The semidirect product Cn G is also known as the wreath product of C and the group G considered as a permutation group on G/H. For any extension L of K with Galois group C, the Galois group of the Galois closure of L over Q is a subgroup of this wreath product. The proposition asserts that the wreath product itself (i.e., the maximal subgroup, which can be viewed as the ‘generic’ case), actually occurs for some L. We give a self-contained proof, but the result also follows from [14, Thm. IV.2.2]; or, for C of order 3 one can use the existence of a generic polynomial for C and apply [7, Prop. 13.8.2].

Proof of Proposition 9.1 Let p = 2 be a prime that is totally split in N and denote by p1, ..., pn the primes in K lying above p. There exists a Galois extension K/K with Galois 7 Page 12 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

group C in which the prime p1 is inert, while p2, ..., pn are totally split (this follows, e.g., from the Grunwald-Wang theorem). Let X be the set of ﬁeld homomorphisms from K to N.SinceG acts on N,wegetan action of G on X by composition. This action is transitive and the stabilizer of the inclusion map ι ∈ X is H,soX is isomorphic to G/H as a G-set. For each σ ∈ X we now consider Nσ := K ⊗K,σ N, where N is viewed as a K-algebra through σ : K → N.TheC-action on K induces a C-action on Nσ by N-algebra automorphisms. Setting Pσ to be the set of primes of N that contain σ (p1), we see that Nσ is a Galois extension of N with Galois group C for which the primes in Pσ are inert and all other primes of N over p are totally split. The G-action on the set of primes of N over p is free and transitive, and Pι consists of

asingleH-orbit: the primes over p1.SincePgσ = gPσ it follows that as σ ranges over X the sets Pσ form a disjoint family. One deduces that the fields Nσ form a linearly disjoint family of C-extensions of N and that the tensor product M = Nσ σ ∈X N N σ ∈ N n = over of all σ with X is a field which is Galois over with Galois group C σ ∈X C. For g ∈ G and σ ∈ X there is a natural field isomorphism gσ : Nσ → Ngσ given by x ⊗ y → x ⊗ gy that extends the map N → N given by y → gy. Combining these maps for all σ ∈ X we obtain an automorphism of the tensor product M that permutes the factors of the tensor product by the g-action on X. Thus, we have extended the G-action on N to a G-action on M. Since each gσ is C-equivariant, the subgroup of Aut (M) generated by G and Cn is the semidirect product Cn G. As the cardinality of this group is the field degree of M over Q we see that M is a Galois extension of Q with Galois group Cn G, and that K is the invariant field of Cn H.

10 Characterization of number ﬁelds Using the previous section we prove a stronger version of Proposition 8.1 for number ﬁelds.

Theorem 10.1 Let K be a number ﬁeld and let k ≥ 3. Then there exists a character q ab q ab χ ∈ GK of order k such that every number ﬁeld L for which there is a character χ ∈ GL

with LL(χ ) = LK(χ) is isomorphic to K.

We will use the following basic facts about Artin L-series of representations of the absolute Galois group GK := Gal(Q/K) for a number ﬁeld K within a ﬁxed algebraic closure Q of Q.

Lemma 10.2 (a) For any two representations ρ and ρ of GQ,LQ(ρ) = LQ(ρ ) is equiv- ∼ alent to ρ −→ ρ . q GQ χ ∈ ab K χ = Q χ (b) For GK ,wehaveL ( ) L (IndGK ( )). q ab q ab (c) For any two number ﬁelds K and L within Q and characters χ ∈ GK and χ ∈ GL

with LK(χ) = LL(χ ) we have an isomorphism of representations of GQ GQ χ −→∼ GQ χ IndGK ( ) IndGL ( )

and the ﬁxed ﬁelds Kχ of χ and Lχ of χ havethesamenormalclosureoverQ. Cornelissen et al. Res. Number Theory (2019) 5:7 Page 13 of 15 7

Proof Fact (a) follows from Chebotarev’s density theorem, comparing Euler factors. The basic fact (b) is due to Artin, see [16, VII.10.4.(iv)]. The isomorphism in (c) follows from (a) and (b). The last statement follows from the fact that the normal closure of Kχ over Q GQ χ is the ﬁxed ﬁeld of the kernel of the representation IndGK ( ).

By a monomial structure of a representation ρ of a group G we mean a G-set L consisting ρ ∈ L ∈ ∈ L of 1-dimensional subspaces of that is G-stable (i.e. gL for all g G and L ), ρ = and such that as a vector space we have L∈L L. By choosing a single nonzero vector of L for each L ∈ L one obtains a basis of ρ such that for every g ∈ G the action of g on ρ is given by a matrix with exactly one non-zero element in each row and in each column. If H is a subgroup of G and χ a linear character of H, then the induced representation ρ = G χ L / IndH ( ) naturally produces a monomial structure that is isomorphic to G H as a G-set.

q ab q ab Proof of Theorem 10.1 If LL(χ ) = LK(χ) for two characters χ ∈ GK , χ ∈ GL ,the GQ χ χ lemma implies that IndGK ( ) has two monomial structures, one arising from and one from χ .WeseethatK and L are isomorphic as number fields if and only if these two monomial structures are isomorphic as GQ-sets (note that they are transitive GQ-sets). In order to prove Theorem 10.1 it therefore suffices to choose χ in such a way that the GQ χ representation IndGK ( )onlyhasasingle monomial structure. In order to find such a character χ we apply Proposition 9.1 to any finite Galois N/Q containing K, and where C =ζ is the subgroup of C× generated by ζ = exp(2πi/k). We let n, G, H,andCn G be as in the proposition, and get an extension M of K within Q with Galois group Gal(M/Q) = Cn G. We order the coordinates of Cn in such a way that the action of H on Cn fixes the first coordinate, so the map

n × Gal(M/K) = C H → C :(a1, ...,an,h) → a1

q ab is a group homomorphism, and extends to a character χ ∈ GK . The induced represen- ρ = GQ χ M/Q = n tation IndGK ( ) factors over Gal( ) C G and it comes with a monomial n structure L ={L1, ...,Ln} such that each element (a1, ...,an) ∈ C acts on Li as scalar n multiplication by ai. It follows that L is exactly the set of 1-dimensional C -submodules of ρ, the so-called character eigenspaces for the action of Cn on ρ. To ﬁnish the proof we will show that L is the unique monomial structure on the representation ρ. Suppose that M is another monomial structure on ρ. The trace of the element c = (ζ , 1, ..., 1) ∈ Cn on ρ is equal to n − 1 + ζ . On the other hand, c permutes the elements of M , so the trace of c on ρ is also equal to the sum of k-th roots of unity

ζM ∈ μk where M ranges over those lines M ∈ M with cM = M,andζM is the scalar by which c then acts on M.Sincek ≥ 3andL and M have the same number of elements, Lemma 6.6 implies that cM = M for all M ∈ M . It follows that c acts trivially on the set M .SincetheG-conjugates of c generate Cn we deduce that Cn acts trivially on the set M .Thus,M consists of 1-dimensional Cn-submodules of ρ.ThisimpliesthatM ⊂ L , so M = L for cardinality reasons.

Remark 10.3 Not every representation has a unique monomial structure: consider the

isometry group of a square, the dihedral group D4 of order 8, with its standard 2- dimensional representation. It has two distinct monomial structures (consisting of the

axes and the diagonals) and these are not isomorphic as D4-sets. 7 Page 14 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7

By realizing D4 as a Galois group over Q this gives rise to quadratic fields K and and q ab q ab L with quadratic characters χ ∈ GK [2] and χ ∈ GL [2] that satisfy LK(χ) = LL(χ ) K L while is not isomorphic to . This shows that the method√ of proof of the theorem fails ≥ Q 4 /Q −→∼ without the assumption that √k 3. Concretely,√ Gal( ( 2,i) ) D4, and we find χ = χ K = Q L = Q χ χ LK( ) √LL( ) for √( 2), √ (i 2) and and uniquely determined by K = Q 4 L = Q + 4 χ ( 2) and χ (i 2, (1 i) 2). √ In [18, Thm. 3.2.2] it is shown that K = Q( 8 5) provides a counterexample to the statement of the theorem for k = 2. On the other hand, in [18, Thm. 2.2.2], a similar method as in our proof is used to show that every number field is characterized uniquely by the L-series of two suitable quadratic characters.

11 Comparison of different methods of proof There is an interesting “incompatibility of proof techniques” between the case of global function fields and that of number fields. Namely, the approach of the proof of Proposi- tion 8.1 for function fields does not transfer in an obvious way to number fields. Indeed, for K = K∗ · O∗ O∗ a number field , ker(recK) K,+, where K,+ is the closure of the totally positive units (i.e., units of OK that are positive in every real embedding of K) in the finite ideles; this follows from the description of the connected component of the idele class group by Artin [2, Ch. IX]. Hence the method of proof of 8.1 transferred literally to number fields yields the weaker conclusion that

K∗ · O∗ = L∗ · O∗ ( K,+) L,+.

It is unclear to us whether one can deduce that (K∗) = L∗ from the conditions in Theorem 3.1. The issue is similar to the one raised in [12, 3.3.2]. On the other hand, it is not possible to copy the proof of Theorem 10.1 for function

fields, as this would force fixing a rational subfield Fq(t)insidebothK and L (that plays the role of Q in the number field proof), for which there are infinitely many, non-canonical, choices. However, Theorem 10.1 does hold in the relative setting of separable geometric extensions of a fixed rational function field of characteristic not equal to 2, compare [19]. It is unclear to us whether the analogue of Theorem 10.1 holds for a global function field without fixing a rational subfield. It does seem that L-series of global function fields, as polynomials in q−s, contain less arithmetical information than their number field cousins (compare [11]).

Author details 1Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands, 2Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands, 3School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK, 4Mathematics Department, Caltech, Mail Code 253-37, 1200 E. California Blvd., Pasadena, CA 91125, USA, 5Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada, 6Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Toronto, ON M5S 2E4, Canada.

Received: 24 April 2018 Accepted: 10 November 2018 Published online: 26 November 2018

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